Is it possible to have a Group without any elements inside it?

Question

I know that any set with just one single element forms a group. The single element satisfies all the axioms. However, similarly to how we have an empty set, which is nevertheless still a set, is it possible to have a sort of "empty" group that would "vacuously" satisfy the group axioms?

I have looked at the Wikipedia page for Group, however I could not find an explicit statement that said that a "0" group can not exist.

If an "empty" group does not in fact exist, then I am confused as to why an "empty" category can exist. It would seem like sometimes you can in fact have "empty" trival objects, and then in other areas of mathematics you can not. I would really appreciate an intuitive discussion as to why that could be?

Thank you so much!

Answer 1

An empty group cannot exist, because one of the group axioms explicitly asks for the existence of an identity element.

Answer 2

No, by definition a group must contain an identity element.